# Point Slope Form

Slope of a line represents the Ratio of change in ‘Y’ to change in ‘X’ between two points on a line, also known as gradient of line. If the Slope of line is undefined or not defined then it is said to be a Slope of a vertical line is undefined whereas slope of a horizontal  line is always zero.
The point Slope of an equation of line is given as
y - y$_1$ = m (x - x$_1$)
or
m = (x - x$_1$) / (y – y$_1$)
Here x$_1$ and y$_1$ are the given point in the equation ‘m’ is the slope of a line equation and (x, y) is any point located on the line. The subscripts with points ‘x’ and ‘y’ show that we are working with two points of a linear equation. It completely depends on us which subscript we want to use as first or which one as a second. The only thing matter is that we must subtract x’s and y’s in same order.
Generally in such problems of finding slope of a linear equation we only have an equation. To find out its slope we first need to graph the line on the plane so that we can get two points by using them we can find out slope of the equation.

## Standard Form

The point slope form of a straight line is the equation of line having slope '$m$' and passing through the point ($x_{1}$, $y_{1}$).

Equation for point-slope form is $y - y_{1}$ = $m(x - x_{1})$

where, $(x_{1},\ y_{1})$: Known point.

m: Slope of the line.

$(x,\ y)$: Other point on the line.

## Convert Point Slope to Slope Intercept

To understand the behavior of a line equation, we use Slope Intercept Form. So, now we discuss how we convert Point slope to Slope intercept:
We use following steps for evaluating the Slope intercepts form from point slope -
Step 1: First of all, we check that point slope is given or not and if point slope is not given in question, then we calculate slope of given point (x, y) by following formula -
Slope = m = y / x,
Like Slope of a line which passes through (1, 2), then slope is 2/1 = 2.
Step 2: After evaluation of slope, we calculate slope intercepts form by using following formula -
y - y_1 = m (x - x_1),

Step 3: Convert into slope intercept form
The form of an equation of a line given above is similar to the equation of the Straight Line. Basically in the point slope form the points (x$_1$, y$_1$) are actually the points (0, c) for the straight line equation.
Both forms can be interchanged with one another. Suppose we have
y - y$_1$ = m (x - x$_1$),
On putting the coordinates (0, b) in spite of (x$_1$, y$_1$), we get
y - b = m (x - 0),
y - b = mx,
y = mx + b, which is required form.

## Linear Equations in Point Slope Form

There are two main aspects to a linear equation: The slope and the y intercept. The slope of a line refers to the rate of change of the dependent variable with respect to independent variable. Normally the independent variable is ‘$x$’ and the dependent variable is ‘$y$’.

The equation of a line that we saw above is normally called a linear equation in standard form or general form. This equation would be of the form:

$ax\ +\ by\ +\ c$ = $0$

Another form of equation of line that we saw above was the slope intercept form. It is like this:

$y$ = $mx\ +\ b$

Here, the $m$ refers to the slope of the line. The slope is simply rise over run.

$slope$ = $\frac{rise}{run}$

Slope of a line plays an important role when we represent the linear equation algebraically and graphically. While plotting a graph, the number of units that the graph moves up for every unit that it move to the right is rise over run.
If we consider any two points on the graph of a linear equation, $(x_{1},\ y_{1})$ and $(x_{2},\ y_{2})$. Then the slope of the line can be calculated using the formula:

$slope$ = $m$ = $\frac{rise}{run}$ = $\frac{(y_{2}\ -\ y_{1})}{(x_{2}\ -\ x_{1})}$

Thus if we know the coordinates of any two points that lie on a line, then we can find its slope. We shall see in a bit that when we know the coordinates of any two points on a line, we can also find the equation of the line.

The $b$ in the slope intercept form refers to the $y$-intercept. The $y$ intercept is the point on the $y$ axis where the graph of the linear equation cuts the $y$ axis.

## Examples

Some examples on the topic are illustrated below:
Example 1
:  The equation of the line that passes through the point (-3,1) with a slope of 2.
Solution:  Here m = 2 and x_1 = -3 and y_1 = 1
Substituting all the given values to the standard form, the equation is
y - y$_1$ = m (x - x$_1$),
y - 1 = 2(x - (-3)
y - 1 = 2(x + 3)

Example 2: Write a slope intercepts form for a line having slope -1 and which passes through a point (2, -3)?
Solution
Step 1: Given, line passes through a point (2, -3) and Slope = m = -1
Step 2: Write a equation of line which passes through the point (2, -3) and m = -1
line is passes y - y$_1$ = m (x - x$_1$),
y - (-3) = (-1) (x - 2),
y + 3 = -x + 2
y + x = -1
So, intercepts form of a given line is y + x = -1 or x + y  = -1.